2 Put LaTeX tags around equation

The chain rule for differentiation

$D(f \circ g ) = Df \circ Dg Dg$

is the first example of functoriality one meets and counts as analyis I guess! Of course to properly interpret this it is best to think of f and g as maps between manifolds and Df and Dg as their tangent maps defined on the tangentbundles of the manifolds. The functor of differentiation is thus the functor of taking the tangent bundle (on objects) and tangent maps (on functions).